Modeling Drinking Behavior Progression in Youth: a Non-identified Probability Discrete Event System Using Cross-sectional Data

Abstract

Background

The probabilistic discrete event systems (PDES) method provides a promising approach to study dynamics of underage drinking using cross-sectional data. However, the utility of this approach is often limited because the constructed PDES model is often non-identifiable. The purpose of the current study is to attempt a new method to solve the model.

Methods

A PDES-based model of alcohol use behavior was developed with four progression stages (never-drinkers [ND], light/moderate-drinker [LMD], heavy-drinker [HD], and ex-drinker [XD]) linked with 13 possible transition paths. We tested the proposed model with data for participants aged 12–21 from the 2012 National Survey on Drug Use and Health (NSDUH). The Moore-Penrose (M-P) generalized inverse matrix method was applied to solve the proposed model.

Results

Annual transitional probabilities by age groups for the 13 drinking progression pathways were successfully estimated with the M-P generalized inverse matrix approach. Result from our analysis indicates an inverse “J” shape curve characterizing pattern of experimental use of alcohol from adolescence to young adulthood. We also observed a dramatic increase for the initiation of LMD and HD after age 18 and a sharp decline in quitting light and heavy drinking.

Conclusion

Our findings are consistent with the developmental perspective regarding the dynamics of underage drinking, demonstrating the utility of the M-P method in obtaining a unique solution for the partially-observed PDES drinking behavior model. The M-P approach we tested in this study will facilitate the use of the PDES approach to examine many health behaviors with the widely available cross-sectional data.

1. INTRODUCTION

Underage drinking, or alcohol use by individuals less than 21 years of age, is associated with considerable social and health burdens to adolescents, their families, and the society [1–3]. Conventional epidemiological approaches, such as normative trajectory model and multiple trajectory models have been commonly used to study the course of alcohol use over the periods from adolescence to young adulthood. The establishment of these approaches usually demands the availability of longitudinal surveys to follow changes in alcohol use behavior [4]. With longitudinal surveys, subjects are closely tracked over time, thereby allowing direct investigation of drinking onset, variable drinking levels, and their temporal relationship with risk factors and health consequences. Despite numerous strengths, the conduct of longitudinal studies can be technically demanding (e.g., sophisticated design for long-term follow-up) and practically challenging (e.g., high costs for implementation and high burdens for both investigators and participants) [5]. Therefore, longitudinal data are not widely available. Among the longitudinal data that are available for use, many are limited in sample size and lack of generalizability to larger populations. Even if “good” longitudinal data are available, there are limitations to such data. The most obvious issues include but are not limited to potential selection bias, loss to follow-up, Hawthorne effect related to repeated assessments, and period and cohort effect inherited with such data, which may confound the investigation of an HIV risk behavior, such as underage drinking [6–9].

Cross-sectional data are widely available from a number of sources, such as the National Survey Drug Use and Health (NSDUH), Behavioral Risk Factor Surveillance System (BRFSS), and Monitoring the Future (MTF) studies. Although individual participants are not tracked over time in a cross-sectional survey, cross-sectional data may contain longitudinal information to support the investigation of dynamic changes in underage drinking under certain conditions [10, 11]. For instance, if we assume that changes in a behavior at the population level in the same age group over a short period (e.g., one year or shorter) are relatively small compared with changes in the same behavior across two consecutive age groups, a cross-sectional survey can be considered equivalent to a longitudinal survey with two waves of data collection for subjects aged (a to n−1) at wave one and aged (a+1 to n) at wave two. In this case, the number of persons in different stages of the drinking behavior across two consecutive age groups in a cross-sectional survey will be a good approximation of those collected through a two-wave longitudinal survey [12].

Recently, Lin and Chen proposed a novel method that is capable of extracting longitudinal information from cross-sectional data to quantify population dynamics of health behavior [10–12]. This approach is developed based on the Probabilistic Discrete Event System (PDES), an established technique widely used in manufacturing and system engineering [13–16]. According to the PDES theory, given a stable system (e.g., assembly workstation in manufacturing), a cross-sectional survey is a “snapshot” of the whole system. When the system is running normally, data from a “snapshot” is adequate to describe the dynamics of the whole system. In another word, only one wave of cross-sectional data is needed to quantify the dynamics of a stable system if the PDES modeling approach is adopted [10–12]. The PDES method has been successfully applied to model smoking behavior among US teenagers and young adults [10]. In that study, transitions in adolescent smoking were divided into 5 discrete stages/states (e.g., never-smoker, experimenter, self-stopper, regular smoker, and quitter) linked by 11 transitional paths. Transitional probabilities for individual transitional paths between different stages (e.g., from being a never-smoker to be an experimenter) were estimated to quantify the changing smoking behavior across age groups.

Despite several strengths, the established PDES modeling has one limitation: the constructed system cannot be solved without the addition of extra behavioral states. For example, when modeling smoking behavior, Lin and Chen had to define extra states such as “old self-stopper” (e.g., those who stopped smoking one year ago) to form a full-rank matrix in order to obtain a unique solution for the 11 transitional probabilities in the full 5-stage PDES model [10]. Such extra states are often impractical to define with existing survey data. Even if data are available, such data may often be error prone. This is because collecting such data needs intensive recall of the detailed changes in a behavior in the long past. According to the advanced matrix algebra, this limitation can be eliminated by the application of the Moore-Penrose (M-P) generalized matrix method [17].

The PDES behavioral progression model proposed by Lin and Chen [10] can be mathematically presented as a linear system Aσ = b and its general solution is σ = A⁻b, where A⁻ is called the generalized-inverse of coefficient matrix of A. If A⁻ were not unique, the solution σ = A⁻b would not be unique, suggesting the lack of solution to the PDES model. For any linear systems that lack a unique solution, an M-P generalized matrix method can be used to provide a “best-fit” solution [17]. According to the M-P generalized matrix method, for any coefficient matrix A, there is precisely one matrix A⁻ if it satisfies the following four criteria: 1) AA⁻A = A, 2) A⁻AA⁻ = A⁻, 3) (AA⁻)′ = AA⁻, 4) (A⁻A)′ = A⁻A [18, 19]. This one A⁻ is often called Moore-Penrose pseudoinverse and typically denoted as A⁺. With the unique A⁺ identified, the PDES model becomes solvable without the need to identify extra behavioral states. This appealing property of M-P generalized matrix method relaxes the limitation of PDES modeling method in investigating behavioral dynamics.

In this study, we illustrated the application of the M-P generalized inverse matrix method with PDES modeling to model the dynamics of underage drinking among a sample of youth aged 12–21 years in the United States.

2. MATERIALS AND METHOD

2.1. Data and Study Subjects

Data used for this study were derived from the 2012 National Survey on Drug Use and Health (NSDUH). The NSDUH is an annual national survey sponsored by the U. S. Department of Health and Human Services (DHHS), Substance Abuse and Mental Health Services Administration (SAMHSA), and conducted by the Research Triangle Institute, North Carolina. It applies a stratified, multiple-stage random sampling design to provide national estimates of use of tobacco, alcohol and illicit drug for the non-institutionalized U.S. civilian population 12 years of age and older [20]. The trained interviewers administer the survey using a combination of computer-assisted personal interview (CAPI) and audio computer-assisted self-interview (ACASI). Questions of alcohol consumption, illicit drug use and other sensitive behaviors are administered through ACASI to increase the honesty and reliability of responses. More details of the survey design and implementation can be seen from official technical reports [20] or other related studies [21]. The NSDUH data for participants aged 12 to 21 years were used because (1) data provided from this survey are adequate to model drinking behavior across the adolescence-young adulthood time period; and (2) data by single year of age up to age 21 are available for PDES modeling.

2.2. A Proposed PDES Drinking Behavior Progression Model

To construct the PDES drinking behavior model, four progression states for underage alcohol use were defined in part according to the National Institute on Alcohol Abuse and Alcoholism (NIAAA) definition for heavy drinking.

1. Never-Drinker (ND): an individual who has never drunk any alcoholic beverage up to the time of the interview.

2. Light-and Moderate-Drinker (LMD): an individual who drank lightly or moderately in the past 30 days. For men, no more than 4 drinks per day AND no more than 14 drinks per week; for women, no more than 3 drinks per day AND no more than 7 drinks per week [22].

3. Heavy-Drinker (HD): an individual who drank in the past 30 days at levels exceeding single-day or weekly limit listed above or engages in any binge drinking at least once per week [22].

4. Ex-Drinker (XD): an individual with previous alcohol consumption who reports no alcohol consumption in the past month.

With NSDUH data, a participant was defined as a ND if he or she responded negatively to the question: “Have you ever had a drink of any type of alcoholic beverage?” Its state probability was thus computed as the ratio of the number of NDs over the total sample.

For those who ever drank, the frequency of alcohol consumption was assessed by asking “During the past 30 days, how many days did you drink one or more drinks of an alcoholic beverage?” and the quantity was assessed by asking “On the days you drink in the past 30 days, how many drinks do you usually have?” The average weekly alcohol consumption was obtained by multiplying the answers to the frequency and quantity and dividing it by 4.29 (30 days equal to 4.29 weeks). A participant was classified as a HD if reported exceeding the recommended limits or binge drinking: >14 drinks per week or >4 drinks per day for males; >7 drinks per week or >3 drinks per day for females. Binge drinking was assessed by asking “during the past 30 days, how many days did you have 5 or more drinks on the same occasion?” Any participant who reported any alcohol use within the past 30 days below the HD criteria was coded as a LMD. Finally, based on the response to the question “How long has it been since you last drank an alcoholic beverage?” a XD was defined if participants indicated their last drink was “more than 30 days ago”.

With these defined states, we proposed a 13-path PDES drinking behavior progression model (Fig. 1). For example, within a one year period, a ND has a chance of σ₁ to remain in the ND state, σ₂ to become a LMD, σ₃ to become a XD, and σ₄ to become a HD. Similarly, a LMD can quit drinking to be a XD or escalate levels of alcohol consumption to be a HD with the annual probabilities of σ₆ and σ₈, respectively. A heavy drinker has a probability of σ₁₀ to remain drinking heavily, σ₉ to cut down to be a LMD, and σ₁₁ to quit within a year. Finally, a XD can remain abstinent with a probability of σ₁₃ or relapse to light/moderate drinking or heavy drinking with one-year probabilities of σ₇ and σ₁₂, respectively.

Fig. 1.

PDES model of the progression of drinking behavior by states. ND: never-drinker, LMD: light-or moderate-drinker, HD: heavy-drinker, and XD: ex-drinker. σ_i(i=1, 2…13): transitional probabilities from one stage to another.

The proposed PDES drinking behavior progression model in Fig. (1) can also be described mathematically as:

$$G=(Q,\mathit{\sum },\delta ,q_0)$$ (1)

where Q= (ND, LMD, HD, XD) denotes a set of drinking states and Σ = (σ₁, σ₂, …, σ₁₂) denotes a set of events that occurs and leads to the transitions from one state to another. For example, σ₂ represents an event that a ND becomes a LMD within a year. δ:Q×Σ → Q represents a set of transition functions describing what event can occur at what state and the resulting state. For example, in Fig. (1), the transition from ND to LMD can be described as δ(ND, σ₂) = LMD. q₀ indicates the initial state and we use q₀ = ND in our PDES drinking behavior progression model. To simplify the notation, we use q_i where q_i ∈ Q to denote the state probability of one state in Q and use σ_i to denote the probability that a transitional event σ_i occurs. For instance, LMD(a) indicates the probability of being a LMD at age a; σ₂(a) indicates the probability of the onset of moderate drinking at age a.

2.3. Methods to Solve the PDES Drinking Behavior Progression Model

2.3.1. Matrix Form of PDES Drinking Behavior Progression Model

To solve the proposed PDES system, we expressed the model in Fig. (1) with the following four equations:

$$ND(a+1)=ND(a)-ND(a){\sigma }_2(a)-ND(a){\sigma }_3(a)-ND(a){\sigma }_4(a)$$ (2)

$$\mathit{LMD}(a+1)=\mathit{LMD}(a)+ND(a){\sigma }_2(a)+XD(a){\sigma }_7(a)+HD(a){\sigma }_9(a)-\mathit{LMD}(a){\sigma }_6(a)-\mathit{LMD}(a){\sigma }_8(a)$$ (3)

$$HD(a+1)=HD(a)+\mathit{LMD}(a){\sigma }_8(a)+XD(a){\sigma }_{12}(a)+ND(a){\sigma }_4(a)-HD(a){\sigma }_9(a)-HD(a){\sigma }_{11}(a)$$ (4)

$$XD(a+1)=XD(a)+ND(a){\sigma }_3(a)+\mathit{LMD}(a){\sigma }_6(a)+HD(a){\sigma }_{11}(a)-XD(a){\sigma }_7(a)-XD(a){\sigma }_{12}(a)$$ (5)

Each equation represents the transitional relationship of one drinking state at age a+1 with other states and events at age a. For instance, the state probability of never-drinkers at age a+1 (ND(a+1)) equals to the total number of ND at age a minus the number of those who initiate drinking (LMD + XD+HD) within one year, and divided by the total number of ND at age a. Similar explanations can be made for the other equations.

To solve the constructed model, the following four equations were added to reflect the relationships among the transitional probabilities for each of the four stages:

$${\sigma }_1(a)+{\sigma }_2(a)+{\sigma }_3(a)+{\sigma }_4(a)=1$$ (6)

$${\sigma }_5(a)+{\sigma }_6(a)+{\sigma }_8(a)=1$$ (7)

$${\sigma }_9(a)+{\sigma }_{10}(a)+{\sigma }_{11}(a)=1$$ (8)

$${\sigma }_7(a)+{\sigma }_{12}(a)+{\sigma }_{13}(a)=1$$ (9)

Put equations (2) to (9) in a matrix format (10):

$$\left[\begin{array}{ccccccccccccc}0& -ND(a)& -ND(a)& -ND(a)& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& ND(a)& 0& 0& 0& -\mathit{LMD}(a)& XD(a)& -\mathit{LMD}(a)& HD(a)& 0& 0& 0& 0\\ 0& 0& 0& ND(a)& 0& 0& 0& \mathit{LMD}(a)& -HD(a)& 0& -HD(a)& XD(a)& 0\\ 0& 0& ND(a)& 0& 0& \mathit{LMD}(a)& -XD(a)& 0& 0& 0& HD(a)& -XD(a)& 0\\ 1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 1& 1\end{array}\right]\left[\begin{array}{c}{\sigma }_1(a)\\ {\sigma }_2(a)\\ {\sigma }_3(a)\\ {\sigma }_4(a)\\ {\sigma }_5(a)\\ {\sigma }_6(a)\\ {\sigma }_7(a)\\ {\sigma }_8(a)\\ {\sigma }_9(a)\\ {\sigma }_{10}(a)\\ {\sigma }_{11}(a)\\ {\sigma }_{12}(a)\\ {\sigma }_{13}(a)\end{array}\right]=\left[\begin{array}{c}ND(a+1)-ND(a)\\ \mathit{LMD}(a+1)-\mathit{LMD}(a)\\ HD(a+1)-HD(a)\\ XD(a+1)-XD(a)\\ 1\\ 1\\ 1\\ 1\end{array}\right]$$

Equation (10) can be simplified as Aσ = B where A is the coefficient matrix, σ is the solution vector and B is the right-hand side vector.

2.3.2. Estimate the State Probabilities for ND, LMD, HD and XD

To solve for the PDES model Aσ = B, state probabilities for ND, LMD, HD and XD were estimated to construct coefficient matrix A and vector B. SAS 9.4 was used to derive state probabilities with 2012 NSDUH data. Survey weights and strata were included in analyses to account for complex multiple-level sampling. These state probabilities (Table 1) were used as input data for the PDES model to derive transitional probabilities between drinking states. In addition to the total sample, the state probabilities were computed for male and female participants respectively. Since the gender differences were rather small, no gender-specific modeling analysis was conducted.

Table 1.

State probability of subjects in the four defined drinking states, aged 12–21, United States, 2012 NSDUH data.

Age	Total/Gender	ND*	LMD†	HD‡	XD§
12	Total	0.9335	0.0098	0.0011	0.0557
	Male	0.9300	0.0103	0.0015	0.0582
	Female	0.9370	0.0092	0.0007	0.0531
13	Total	0.8739	0.0217	0.0072	0.0972
	Male	0.8678	0.0256	0.0040	0.1025
	Female	0.8801	0.0177	0.0104	0.0918
14	Total	0.7410	0.0656	0.0237	0.1697
	Male	0.7393	0.0673	0.0208	0.1726
	Female	0.7430	0.0637	0.0268	0.1665
15	Total	0.6235	0.0796	0.0480	0.2489
	Male	0.6390	0.0768	0.0474	0.2368
	Female	0.6072	0.0826	0.0486	0.2616
16	Total	0.5107	0.1096	0.0863	0.2934
	Male	0.5230	0.0953	0.0913	0.2904
	Female	0.4974	0.1250	0.0809	0.2967
17	Total	0.4033	0.1308	0.1444	0.3214
	Male	0.4016	0.1320	0.1419	0.3245
	Female	0.4051	0.1297	0.1469	0.3183
18	Total	0.3136	0.1951	0.1997	0.2916
	Male	0.3133	0.2052	0.2026	0.2789
	Female	0.3140	0.1840	0.1965	0.3055
19	Total	0.2587	0.2001	0.2341	0.3071
	Male	0.2680	0.1792	0.2601	0.2927
	Female	0.2488	0.2224	0.2064	0.3224
20	Total	0.1901	0.2401	0.2853	0.2844
	Male	0.1855	0.2059	0.3191	0.2896
	Female	0.1950	0.2757	0.2502	0.2791
21	Total	0.1198	0.3503	0.3414	0.1885
	Male	0.1093	0.3504	0.3747	0.1657
	Female	0.1305	0.3503	0.3075	0.2117

^*ND: never-drinker.

^†LMD: light/moderate-drinker.

^‡HD: heavy-drinker.

^§XD: ex-drinker.

2.3.3. M-P Solution to the PDES Drinking Progression Model

Since the matrix equation (10) is not full-ranked: the number of unique equations (n=7) is smaller than the number of unknowns (n=13), and thereby there is no unique solution for the PDES model Aσ = B if conventional matrix algebra is utilized. We replaced the conventional inverse matrix solution with the M-P generalized inverse matrix method, which is devised to identify a unique solution for σ using the minimum distance criteria [17, 23, 24].

We used the R-3.1.1 statistical software for the development and solution for the PDES model of underage drinking behavior. Since the transitional probabilities by definition must be nonnegative, we imposed the condition σ_i ≥ 0 to solve for σ₁, σ₂, …, σ₁₃. The R programs of the current paper are included in the Appendix.

3. RESULTS

A total of 26,704 participants aged 12–21 from the 2012 NSDUH were included for our analyses. Table 1 describes the state probabilities of ND, LMD, HD, and XD by single year of age group from 12 to 21 years old, overall and by gender. For instance, at age 12, 93.4% of adolescents were ND, 1% were LMD, 0.1% were HD, and 5.6% were XD, respectively. The state probability for ND declined while state probabilities for LMD and HD increased with age.

Table 2 summarizes the 13 estimated transitional probabilities from the PDES drinking progression model solved by the M-P generalized inverse matrix method. According to the 2012 NSDUH data, a teenager aged 12 who have never used alcohol (ND) in 2012 had a 94% chance (σ₁=0.94) to remain as a ND in a year or otherwise had a 6% chance (1 − σ₁) to initiate alcohol consumption. At age 20, the chance of onset of drinking increased to 47%, indicating almost half of young adults of this age who never used alcohol would begin drinking when approaching the legal drinking age at 21 years old.

Table 2.

Estimated transitional probabilities for the PDES drinking behavior progression model.

Age (Yrs)	σ1	σ2	σ3	σ4	σ5	σ6	σ7	σ8	σ9	σ10	σ11	σ12	σ13
12	0.94	0.00	0.06	0.00	0.32	0.42	0.33	0.25	0.33	0.32	0.34	0.08	0.60
13	0.85	0.02	0.14	0.00	0.35	0.38	0.42	0.26	0.34	0.31	0.35	0.16	0.42
14	0.84	0.00	0.16	0.00	0.33	0.44	0.30	0.23	0.34	0.29	0.37	0.15	0.55
15	0.82	0.00	0.18	0.00	0.32	0.41	0.27	0.28	0.33	0.30	0.37	0.20	0.52
16	0.79	0.00	0.21	0.00	0.30	0.39	0.24	0.31	0.31	0.32	0.37	0.28	0.47
17	0.78	0.03	0.16	0.04	0.32	0.36	0.30	0.32	0.32	0.32	0.36	0.30	0.40
18	0.82	0.00	0.13	0.04	0.30	0.38	0.28	0.32	0.30	0.32	0.38	0.32	0.40
19	0.73	0.06	0.10	0.10	0.31	0.34	0.30	0.35	0.31	0.35	0.35	0.35	0.35
20	0.63	0.17	0.05	0.16	0.39	0.23	0.40	0.38	0.40	0.39	0.22	0.39	0.22

σ₁:ND→ND; σ₂: ND→LMD; σ₃: ND→XD; σ₄: ND→HD; σ₅: LMD→LMD; σ₆: LMD→XD; σ₇: XD→LMD; σ₈: LMD→HD; σ₉: HD→LMD; σ₁₀: HD→HD; σ₁₁: HD→XD; σ₁₂: XD→HD; σ₁₃: XD→XD.

In our PDES model, 3 pathways were related to drinking initiation: never-drinkers to light/moderate-drinkers (σ₂, ND→LMD); never-drinkers to ex-drinkers (σ₃, ND→XD); and never-drinkers to heavy-drinkers (σ₄, ND→HD). Fig. (2) depicts the age pattern of these 3 transitions and the overall probability of onset, respectively. Before age 16, σ₃, the transition from never-drinkers to ex-drinkers dominated the progression of drinking onset. However, starting at the age of 16, the upward trend in σ₃ was replaced by a downturn trend while other two transitions (σ₂ and σ₄) began to play a role. This resulted in a dip in the overall rate of initiation, followed by a sharp increase up to the last age group.

Fig. 2.

Age patterns of the probabilities of the overall and three types of drinking onset, 12 to 20 years old. The overall probabilities of drinking onset (σ₂ + σ₃ + σ₄); σ₃: the probability that never-drinkers (ND) directly progress to ex-drinkers (XD); σ₄: the probability that ND directly progress to heavy-drinkers (HD); σ₂: the probability that ND progress to light/moderate-drinkers (LMD)

Fig. (3) shows the age pattern of the probability for ex-drinkers relapsing to LMD (σ₇, XD->LMD) and HD (σ₁₂, XD->HD), respectively. When adolescents grew older, the probability of relapsing to HD continued to climb up to 39% at age 20, suggesting that 39% of ex-drinkers aged 20 years old would start to drink heavily within one year. On the other hand, the probability of relapsing to LMD showed a “J-shape” trend: the general downturn trend continued until age 16, followed by an upward trend up to age 20.

Fig. 3.

Age patterns of the probabilities for relapse to light/moderate-drinkers (LMD) and heavy-drinkers (HD). σ₇: XD→LMD; σ₁₂: XD→HD.

The age patterns for the two types of quitting for drinking (σ₆, σ₁₁) are presented in Fig. (4). Up to age 16, light/moderate-drinkers were more likely to quit than heavy drinkers. The differences were negligible at age 17. There was a sharp decline in the probabilities to quit for both LMD and HD after 18 years of age.

Fig. 4.

Age patterns of the probabilities for light/moderate-drinkers (LMD) and heavy-drinkers (HD) to quit. σ₆: probability for LMD to quit; σ₁₁: probability for HD to quit.

4. DISCUSSION

In this study, we attempted the M-P generalized inverse matrix method to solve a partially-observed PDES system proposed for modeling underage drinking. Data used for this study are derived from a national random sample survey of individuals aged 12–21 years. Through our research, we demonstrated that the utility of the M-P method in obtaining a unique solution for a partially observed system of the proposed PDES drinking behavior model. The system consists of 13 unknown transitional probabilities to be solved but observed data only allow the construction of 7 independent equations. The modeling results appear reasonable. To our knowledge, this is the first study to test the utility of an advanced mathematical method in solving PDES health behavior model without identifying extra states.

4.1. New Data Describing Drinking Behavior Progression

Our findings from the PDES model demonstrate an increasing age trend of overall drinking onset among individuals from 12 to 21 years old. The transition from never-drinkers to ex-drinkers, or the initiation of XD (σ₃), represents the alcohol use that is irregular, experimental and has not evolved into consistent and regular drinking behavior. We observed an inverse “J” shape curve for the transition to XD (σ₃) from early to late adolescent period. This finding is consistent with the developmental principle of underage drinking. For instance, the developmental maturation varies within different regions of the developing brain [25,26]. The regions governing some emotional and motivational systems mature early in the brain, whereas the systems responsible for cognitive and self-regulatory controls mature more gradually throughout adolescence and young adulthood [27]. This creates a developmental gap that may help explain the early increase in experimental use of alcohol with continuous reduction when growing up. On the other hand, we observed a dramatic increase in the initiation of LMD and HD after age 18 years. These increases may reflect the impact of particular life stages, such as entering college and approaching to the legal drinking age. Although alcohol use usually begins before entry to college, pressure to use or heavy use of alcohol may be intensified when a student is increasingly interacting with peers in college, is getting closer to legal drinking age, and parents are less present. For example, besides the rise in probabilities of initiation of LMD and HD, our results suggest that probabilities of quitting among those with LMD and HD also decreased substantially from age 18 to 20 years old.

4.2. Advantages of PDES-Based Modeling for Health Behavior Research

The PDES model allows the identification and quantification of transitions across various drinking behavior transition states. This level of information cannot be achieved through the conventional prevalence trend analysis. For instance, rate of heavy drinking at certain age a, HD (a), is a static prevalence measure and its increase or decrease is determined by multiple events, such as how likely a heavy-drinker at age (a−1) remains as a heavy drinker, how likely a never-drinker initiates heavy drinking as well as how likely a currently light/moderate-drinker progresses to a heavy drinker over one year period. Better understanding of these underlying changing mechanisms is of particular importance for the development of effective alcohol prevention and control programs. However, such useful information may not be revealed without the PDES model.

Well-validated with longitudinal data [12], PDES method adds an alternative approach to health behavior research by using more available cross-sectional data [3,28–31]. With the PDES model, the transitional probabilities can be estimated with data from one single cross-sectional survey. These probabilities can be used to describe population dynamics of drinking behaviors progression over time. The PDES-based method is also flexible for subgroup analysis that allows the evaluation of potential factors associated with drinking behavior progression.

In addition to describing the behavior dynamics, the PDES method can be used to evaluate the effect of alcohol prevention and control practice. A contrast of transitional probabilities (e.g., probability of drinking onset) between exposed and un-exposed youth will provide data for assessing the effects of early alcohol prevention intervention. Additionally, when multiple-year cross-sectional data are available, a historical analysis comparing transitional probabilities across time can be related to various alcohol prevention activities (e.g., alcohol taxation, school-based program, changing legal drinking age). Other useful information can also be derived when comparing transitional probabilities over time and across interventions: (1) intervention effects on specific drinking progression pathway (e.g., reducing initiation of heavy drinking or increasing quitting); (2) progression pathways that are sensitive to interventions; (3) the amount of change needed in a transitional step in order to achieve a pre-determined drinking prevention objective.

CONCLUSION

The establishment of the PDES model provides a powerful tool to study the longitudinal dynamics of health behavior with cross-sectional data. Previously, this method has been hindered by the lack of fully observed data to solve the constructed models. The findings of this study indicate that the M-P generalized inverse matrix approach is a promising approach to address this challenge. Future research is needed to validate the M-P method and the results of this analysis using other methods with different mechanisms in solving partially observed systems such as the convex optimization [32].

APPDENDIX

A program for the implementation of a PDES

The following R program provides the step-by-step illustration of the implementation of the PDES model with the M-P generalized inverse matrix method.

#step 1: read in data
dat = read.csv(“drinkdata_30d_new.csv”, header=T)
library(limSolve)
d1 <-diag(13)
e1 <-rep(0,13)
PDES1 = function(a){
# get the coefficient matrix in equation 10 (8 equations)
r1 = c(0,-dat$ND[a],-dat$ND[a],-
dat$ND[a],0,0,0,0,0,0,0,0,0)
r2 = c(0,dat$ND[a],0,0,0,-dat$LMD[a],dat$XD[a],-
dat$LMD[a],dat$HD[a],0,0,0,0)
r3 = c(0,0,0,dat$ND[a],0,0,0,dat$LMD[a],-dat$HD[a],0,-
dat$HD[a],dat$XD[a],0)
r4 = c(0,0,dat$ND[a],0,0,dat$LMD[a],-
dat$XD[a],0,0,0,dat$HD[a],-dat$XD[a],0)
r5 = c(1,1,1,1,0,0,0,0,0,0,0,0,0)
r6 = c(0,0,0,0,1,1,0,1,0,0,0,0,0)
r7 = c(0,0,0,0,0,0,0,0,1,1,1,0,0)
r8 = c(0,0,0,0,0,0,1,0,0,0,0,1,1)
out = rbind(r1,r2,r3,r4,r5,r6,r7,r8)
# get the right-side vector
vec = c(dat$ND[a+1]-dat$ND[a],dat$LMD[a+1]-
dat$LMD[a],dat$HD[a+1]-dat$HD[a], dat$XD[a+1]-
dat$XD[a],1,1,1,1)
lsei(A=NULL,B=NULL,E=out,F=vec,G=d1,H=e1)
} # end of PDES1
#calculations to generate Table 2 in the current paper
newtab1=rbind(PDES1(1)$X,PDES1(2)$X,PDES1(3)$X,PD
ES1(4)$X,PDES1(5)$X,PDES1(6)$X,PDES1(7)$X,PDES1(
8)$X,PDES1(9)$X)
newtab1
colnames(newtab1)=paste(“sig”,1:13,sep=“”)
rownames(newtab1)=dat$Age[-10]
print(newtab1)
#output to a csv file
write.csv(newtab1,”new_30d.csv”)